(Leftmost-Outermost) Beta Reduction is Invariant, Indeed

Abstract

Abstract. Slot and van Emde Boas' weak invariance thesis states that reasonable machines can simulate each other within a polynomial overhead in time. Is λ-calculus a reasonable machine? Is there a way to measure the computational complexity of a λ-term? This paper presents the first complete positive answer to this long-standing problem. Moreover, our answer is completely machine-independent and based on a standard notion in the theory of λ-calculus: the length of a leftmost-outermost derivation to normal form is an invariant, i.e. reasonable, cost model. Such a theorem cannot be proved by directly relating λ-calculus with Turing machines or random access machines, because of the size-explosion problem: there are terms that in a linear number of steps produce an exponentially large output. The first step towards the solution is to shift to a notion of evaluation for which the length and the size of the output are linearly related. This is done by adopting the linear substitution calculus (LSC), a calculus of explicit substitutions modeled after linear logic proof nets and admitting a decomposition of leftmost-outermost derivations with the desired property. Thus, the LSC is invariant with respect to, say, random access machines. The second step is to show that the LSC is invariant with respect to the λ-calculus. The size explosion problem seems to imply that this is not possible: having the same notions of normal form, evaluation in the LSC is exponentially longer than in the λ-calculus. We solve such an impasse by introducing a new form of shared normal form and shared reduction, called useful. Useful evaluation produces a compact, shared representation of the normal form, by avoiding those steps that only unshare the output without contributing to β-redexes, i.e. the steps that cause the blow-up in size. The main technical contribution of the paper is indeed the definition of useful reductions and the thorough analysis of their properties. ACM CCS: [Theory of computation]:Models of computation-Computability-Lambda calculus; Models of computation-Abstact machines; Logic-Proof theory / Linear logic / Equational logic and rewriting; Computational complexity and cryptography-Complexity theory and logic.

Tight typings and split bounds

“…The head case is the simplest possible one. The leftmost case is the natural iteration of the head one, and the only known strong strategy whose number of steps provides a reasonable cost model [Accattoli and Dal Lago 2016]. Multi types bounding the lengths of leftmost normalising terms have been also studied in [Kesner and Ventura 2014], but the exact number of steps taken by the leftmost strategy has not been measured via multi types before-therefore, this is a new result, as we now explain.…”

The Negligible and Yet Subtle Cost of Pattern Matching

“…We plan to include them in a longer, journal version of this work. Open terms and evaluation under abstraction instead require more sophisticated machineries [10,9,5,11,12], whose adaptation to CbNeed and pattern matching is under development.…”

“…The work of Accattoli and coauthors [6,9,4] has however repeatedly showed that these tasks require an overhead linear in the number of β-steps and the size of the initial term, and in some cases even logarithmic in the size of the initial term (see the companion paper [7])-in the terminology of this paper, the costs of search and α-renaming are negligible. At the technical level, for the study of cost models we mostly adopt the techniques and the terminology (linear substitution calculus, subterm invariant, harmony, etc) developed by Accattoli and his coauthors (Dal Lago, Barenbaum, Mazza, Sacerdoti Coen, Guerrieri) in [10,6,9,11,4].…”

Efficient Implementation of Evaluation Strategies via Token-Guided Graph Rewriting

“…The challenge here is how to produce a unified framework which is flexible enough to analyse and guide the choices required by this trade-off. Recent studies by Accattoli et al [4,2,1] clearly establish classes of efficiency for a given evaluation strategy. They characterise efficiency by means of the number of beta-reduction applications required by the strategy, and introduce two efficiency classes, namely "efficient" and "reasonable."…”